Weakly nonlocal Poisson brackets

Weakly nonlocal Poisson brackets are a distinguished type of Hamiltonian operators for PDEs which contain nonlocal (integral) terms of a prescribed form. They have been introduced in

• A.Ya. Maltsev, S.P. Novikov: On the local systems Hamiltonian in the weakly nonlocal Poisson brackets, Physica D: Nonlinear Phenomena, 156 (1-2) (2001), 53-80.

An important subclass of the weakly nonlocal Poisson brackets is provided by nonlocal homogeneous first-order Hamiltonian operators defined by metric of constant curvature, first introduced in

• E.V. Ferapontov, O.I. Mokhov, Non-local Hamiltonian operators of hydrodynamic type related to metrics of constant curvature, Uspekhi Math. Nauk 45 no. 3 (1990), 191–192, English translation in Russ. Math. Surv. 45 (1990), 281–219,

and widely generalized to metrics on hypersurfaces of a pseudoeuclidean space in

• E.V. Ferapontov, Hamiltonian systems of hydrodynamic type and their realization on hypersurfaces of a pseudoeuclidean space, Geom. Sbornik 22 (1990), 59–96, English translation in J. Sov. Math. 55 (1991), 1970–1995.
• E.V. Ferapontov, Differential geometry of nonlocal Hamiltonian operators of hydrodynamic type, Funkts. Anal. i Prilozhen. 25 no. 3 (1991), 37–49; English translation in Funct. Anal. Appl. 25 (1991).
• E.V. Ferapontov, Nonlocal Hamiltonian Operators of Hydrodynamic Type: Differential Geometry and Applications, Amer. Math. Soc. Transl. Vol. 170 no. 2 (1995), 33–58.

The calculation of the Jacobi property for weakly nonlocal Poisson brackets was made into an algorithmic process in the recent paper

• M. Casati, P. Lorenzoni, R. Vitolo: Three computational approaches to weakly nonlocal Poisson brackets , Studies in Applied Mathematics 144 no. 4 (2020) 412–448, DOI: 10.1111/sapm.12302, arXiv:1903.08204

In a new paper

• M. Casati, P. Lorenzoni, D. Valeri, R. Vitolo: Weakly nonlocal Poisson brackets: tools, examples, computations, arXiv:2104.13206 three new software packages for the calculation of the Jacobi property for weakly nonlocal operators, following the algorithm of the previous paper, have been written and presented through several examples: the modified KdV equation, the Heisenberg magnet system and a Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) system. The packages are JACOBI, written in Maple, nlPVA, written in Mathematica, and the module cde_weaklynl.red of the CDE package. The Maple and Mathematica packages are also shipped in versions with parallel computing capabilities. All software and examples are released under the terms of the FreeBSD license.

Prospective users can download here all the above software packages and the examples. The authors welcome any question or comment at their respective email addresses (inside the paper).